scholarly journals Computatıonal Algorıthm for the Numerıcal Solutıon of Systems of Volterra Integro-Dıfferentıal Equatıons

2020 ◽  
pp. 66-76
Author(s):  
Falade Kazeem Iyanda ◽  
Tiamiyu Abd`gafar Tunde
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Analytical Approach ◽  
Computational Algorithm ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Linear And Nonlinear ◽  
Variational Iterative Method

In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.

scholarly journals Numerical solution of nonlinear system of integro-differential equations using Chebyshev wavelets

2015 ◽  
Vol 11 (2) ◽  
pp. 15-34
Author(s):  
H. Aminikhah ◽  
S. Hosseini
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Wavelets ◽  
Linear And Nonlinear

Abstract This paper introduces an approach for obtaining the numerical solution of the linear and nonlinear integro-differential equations using Chebyshev wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution. Comparison of the approximate solution with exact solution shows that the used method is effectiveness and practical for classes of linear and nonlinear system of integro-differential equations.

Numerical solution of the general Volterra nth-order integro-differential equations via variational iteration method

2018 ◽  
Vol 13 (02) ◽  
pp. 2050042
Author(s):  
Fernane Khaireddine
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
General Formula ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Numerical Examples ◽  
Variational Iteration ◽  
Linear And Nonlinear

In this paper, we use the variational iteration method (VIM) to construct approximate solutions for the general [Formula: see text]th-order integro-differential equations. We show that his method can be effectively and easily used to solve some classes of linear and nonlinear Volterra integro-differential equations. Finally, some numerical examples with exact solutions are given.

scholarly journals A New Approximate Analytical Method for ODEs

2014 ◽  
Vol 10 (1) ◽  
pp. 19-30
Author(s):  
Hossein Aminikhah
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
New Method ◽  
Good Stability ◽  
Linear And Nonlinear ◽  
Stability And Accuracy ◽  
Theoretical Considerations

Abstract In this paper, we propose a new algorithm for solving ordinary differential equations. We show the superiority of this algorithm by applying the new method for some famous ODEs. Theoretical considerations are discussed. The first He's polynomials have used to reach the exact solution of these problems. This method which has good stability and accuracy properties is useful in deal with linear and nonlinear system of ordinary differential equations.

scholarly journals Analytic and numerical solution for duffing equations

2016 ◽  
Vol 5 (2) ◽  
pp. 115 ◽  
Author(s):  
Majeed AL-Jawary ◽  
Sayl Abd- AL- Razaq
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Duffing Equations ◽  
Linear And Nonlinear ◽  
New Iterative Method

<p>Daftardar Gejji and Hossein Jafari have proposed a new iterative method for solving many of the linear and nonlinear equations namely (DJM). This method proved already the effectiveness in solved many of the ordinary differential equations, partial differential equations and integral equations. The main aim from this paper is to propose the Daftardar-Jafari method (DJM) to solve the Duffing equations and to find the exact solution and numerical solutions. The proposed (DJM) is very effective and reliable, and the solution is obtained in the series form with easily computed components. The software used for the calculations in this study was MATHEMATICA<sup>®</sup> 9.0.</p>

scholarly journals Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Seyyedeh Roodabeh Moosavi Noori ◽  
Nasir Taghizadeh
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Differential Transform Method ◽  
Hybrid Technique ◽  
Transform Method ◽  
Proportional Delays ◽  
Differential Transform ◽  
Computational Work ◽  
Linear And Nonlinear ◽  
First Time

AbstractIn this study, a hybrid technique for improving the differential transform method (DTM), namely the modified differential transform method (MDTM) expressed as a combination of the differential transform method, Laplace transforms, and the Padé approximant (LPDTM) is employed for the first time to ascertain exact solutions of linear and nonlinear pantograph type of differential and Volterra integro-differential equations (DEs and VIDEs) with proportional delays. The advantage of this method is its simple and trusty procedure, it solves the equations straightforward and directly without requiring large computational work, perturbations or linearization, and enlarges the domain of convergence, and leads to the exact solution. Also, to validate the reliability and efficiency of the method, some examples and numerical results are provided.

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

Numerical Solution for a System of Fractional Differential Equations with Applications in Fluid Dynamics and Chemical Engineering

2017 ◽  
Vol 15 (5) ◽  
Author(s):  
Bijil Prakash ◽  
Amit Setia ◽  
Shourya Bose
Keyword(s):  
Fluid Dynamics ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Crucial Role ◽  
Chemical Engineering ◽  
Haar Wavelets ◽  
Fractional Differential

Abstract In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.

scholarly journals Analytical Method of Modelling the Geometric System of Communication Route

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Wladyslaw Koc
Keyword(s):  
Differential Equations ◽  
Analytical Method ◽  
Analytical Approach ◽  
Linear Change ◽  
Linear And Nonlinear ◽  
Transition Curves

The paper presents a new analytical approach to modelling the curvature of a communication route by making use of differential equations. The method makes it possible to identify both linear and nonlinear curvature. It enables us to join curves of the same or opposite signs of curvature. Solutions of problems for linear change of curvature and selected variants of nonlinear curvature in polynomial and trigonometric form were analyzed. A comparison of determined horizontal transition curves was made and examples of negotiating these curves into a geometric system were given.

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